Salimov R. To the theory of the local behavior for the mappings with finite distortion.

The thesis is devoted to the study of mappings with finite distortion in euclidean and in metric spaces. It is established that Q-homeomorphisms by Martio in euclidean spaces are absolute continuous on lines and differentiable a.e. whenever is locally integrable Q. It is formulated a series of conditions on the function Q(x) and boundaries of the domains under which every Q-homeomorphism between domains in metric spaces with measures. admits a continuous or homeomorphic extension to the boundary. In particular, it is proved a generalization and strengthening of the known Gehring-Martio theorem on homeomorphic extending to the boundary quasiconformal mappings between the quasiextremal distance domains. The results can be applied, in particular, to Riemannian manifolds, the Loewner spaces, the groups by Carnot and Heisenberg.